I don't believe so. Pre-Obergfell states were not required to recognize same-sex marriage licenses from other states. Some did, others did not. You could not get married in Massachusetts, for example, and then move to Missouri and file taxes as a married couple. Drivers license reciprocity is the same. States accept them because it's to their benefit to do so not because the federal government says they must. And there are limits. Just because a 14 year old can drive in Kansas doesn't mean they can in California.
Some licenses, like marriage licenses, yes, are required to be recognized under the “full faith & credit” clause. However, the courts have never ruled that clause extends to driver’s licenses, or carry permits.